Search: id:A138036
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%I A138036
%S A138036 0,1,2,0,1,0,2,1,0,1,2,2,0,2,1,0,1,0,0,1,2,0,2,0,0,2,1,1,0,1,1,0,2,1,2,
%T A138036 0,1,2,1,2,0,1,2,0,2,2,1,0,2,1,2,0,1,0,2,0,1,2,0,0,1,2,1,0,2,0,1,0,2,1,
%U A138036 0,0,2,1,2,1,0,1,2,1,0,2,0,1,0,2,1,1,2,0,1,1,2,0,2,1,2,1,0,2,0,1,0,2,0
%N A138036 From a substitution sequence of Veikko Keronen: "From http://south.rotol.ramk.fi/
               ~keranen/Forssa/UsingTheEmptySet.html (.nb) the reader may find examples 
               of computer algebra programs in which we pay special attention to 
               handling the empty set. Indeed, in our study, the empty set (or the 
               empty word) is many times a natural starting point and the meticulous 
               use of it supports us to keep the programs clear and reliable. In 
               many cases the computation also ends with the empty set as a result. 
               For example, the reader may try the following Mathematica program 
               which starts, in a sense, from {{}} and ends with {}"..."Indeed, 
               this computation shows that Abelian squares are unavoidable over 
               3 letters, since every word of length 8 turns out to contain them. 
               Originally, this style of programming arose in discussions with Stephen 
               Wolfram and our program above is a modified version of the example 
               that Wolfram presents in [35, p. 944].".
%C A138036 Mathematica in flatten a sequence truncates the nulls instead of making 
               zeros
%C A138036 of them as I have been doing. I'm entering this code as it didn't come 
               up in a search an appears to be interesting as a sequence that appears 
               to be chaotic.
%D A138036 http://south.rotol.ramk.fi/keranen/ias2002/NewAbelianSquare-FreeDT0L-LanguagesOver4Letters.nb
%D A138036 S. Wolfram. A New Kind of Science. Wolfram Media, 2002.
%F A138036 Mathematica substitution code...see references: Mathematica section.
%e A138036 Triangular form with "nulls":
%e A138036 {{}},
%e A138036 {{0}, {1}, {2}},
%e A138036 {{0, 1}, {0, 2},
%e A138036 {1, 0}, {1, 2}, {2, 0}, {2, 1}},
%e A138036 {{0, 1, 0}, {0, 1, 2}, {0, 2, 0}, {0, 2, 1}, {1, 0, 1}, {1, 0, 2}, {1, 
               2,
%e A138036 0}, {1, 2, 1}, {2, 0, 1}, {2, 0, 2}, {2, 1, 0}, {2, 1, 2}},
%e A138036 {{0, 1,0, 2}, {0, 1, 2, 0}, {0, 1, 2, 1}, {0, 2,
%e A138036 0, 1}, {0, 2, 1, 0}, {0, 2, 1, 2}, {1, 0, 1, 2}, {1, 0, 2, 0}, {1,
%e A138036 0, 2, 1}, {1, 2, 0, 1}, {1, 2, 0, 2}, {1, 2,
%e A138036 1, 0}, {2, 0, 1, 0}, {2, 0, 1, 2}, {2, 0, 2, 1}, {2, 1, 0, 1}, {2,
%e A138036 1, 0, 2}, {2, 1, 2, 0}},
%e A138036 {{0, 1, 0, 2, 0}, {0, 1, 0, 2, 1}, {0, 1, 2, 0, 1}, {0, 1, 2, 0, 2}, 
               {0, 1, 2, 1, 0}, {0, 2, 0,1, 0}, {0, 2, 0, 1, 2}, {0, 2, 1, 0, 1}, 
               {0, 2, 1, 0,2}, {0, 2, 1, 2, 0}, {1, 0,1, 2, 0}, {1, 0, 1, 2, 1}, 
               {1, 0, 2, 0, 1}, {1, 0, 2, 1, 0}, {1, 0, 2, 1, 2}, {1, 2, 0, 1, 0}, 
               {1, 2, 0, 1, 2}, {1, 2, 0, 2, 1}, {1, 2, 1, 0, 1}, {1, 2, 1, 0, 2}, 
               {2, 0,1, 0, 2}, {2, 0, 1, 2, 0}, {2, 0, 1, 2, 1}, {2, 0, 2,1, 0}, 
               {2, 0, 2, 1, 2}, {2,1, 0, 1, 2}, {2, 1, 0, 2, 0}, {2, 1, 0, 2, 1}, 
               {2, 1, 2, 0, 1}, {2, 1, 2, 0, 2}},
%e A138036 {{0, 1, 0, 2, 0, 1}, {0, 1, 0, 2, 1, 0}, {0, 1,0, 2, 1, 2}, {0, 1, 2, 
               0, 1, 0}, {0, 1, 2, 1, 0, 1}, {0, 2, 0, 1, 0, 2}, {0, 2, 0, 1, 2, 
               0}, {0, 2, 0, 1, 2, 1}, {0, 2, 1, 0, 2, 0}, {0, 2, 1, 2, 0, 2}, {1, 
               0, 1, 2, 0, 1}, {1, 0, 1, 2, 0, 2}, {1, 0, 1, 2, 1, 0}, {1, 0, 2, 
               0, 1, 0}, {1, 0, 2, 1, 0, 1}, {1, 2, 0, 1, 2, 1}, {1, 2, 0, 2, 1, 
               2}, {1, 2, 1, 0, 1, 2}, {1, 2, 1, 0, 2, 0}, {1, 2, 1, 0, 2, 1}, {2, 
               0, 1, 0, 2, 0}, {2, 0, 1, 2, 0, 2}, {2, 0, 2, 1, 0, 1}, {2, 0, 2, 
               1, 0, 2}, {2, 0, 2, 1, 2, 0}, {2, 1, 0, 1, 2, 1}, {2, 1, 0, 2, 1, 
               2}, {2, 1, 2, 0, 1, 0}, {2, 1, 2, 0, 1, 2}, {2, 1, 2, 0, 2, 1}}, 
               {{0, 1, 0, 2, 0, 1, 0}, {0,1, 0, 2, 1, 0, 1}, {0, 1, 2, 1, 0, 1, 
               2}, {0, 2, 0, 1, 0, 2, 0}, {0, 2, 0, 1, 2, 0, 2}, {0, 2, 1, 2, 0, 
               2, 1}, {1, 0, 1, 2, 0, 1, 0}, {1, 0, 1, 2, 1, 0, 1}, {1,
%e A138036 0, 2, 0, 1, 0, 2}, {1, 2, 0, 2, 1, 2, 0}, {1, 2, 1, 0, 1, 2, 1}, {1, 
               2, 1, 0, 2, 1, 2}, {2, 0, 1, 0, 2, 0, 1}, {2, 0, 2,1, 0, 2, 0}, {2, 
               0, 2, 1, 2, 0, 2}, {2,1, 0, 1, 2, 1, 0}, {2, 1, 2, 0, 1, 2, 1}, {2, 
               1, 2, 0, 2, 1, 2}},
%e A138036 {}
%t A138036 Clear[a]; a = With[{n = 8, k = 3}, NestList[DeleteCases[Flatten[Map[Table[Append[ 
               #, i - 1], {i, k}] &, # ], 1], {___, u__, v__} /; Sort[{u}] == Sort[{v}]] 
               &, {{}}, n]]; Flatten[a]
%Y A138036 Cf. A007413, A001285.
%Y A138036 Sequence in context: A156667 A110914 A127505 this_sequence A086372 A089650 
               A085513
%Y A138036 Adjacent sequences: A138033 A138034 A138035 this_sequence A138037 A138038 
               A138039
%K A138036 nonn,uned,tabf
%O A138036 1,3
%A A138036 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), May 02 2008

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